What is the domain of the function $$w(x) = \sqrt{x-2} + \sqrt[3]{x-3}~?$$ Express your answer in interval notation.
Note that $\sqrt[3]{x-3}$ is defined for all $x$. The only restriction is due to the $\sqrt{x-2}$ term, which is defined only if $x-2$ is nonnegative. Thus, the domain of $w(x)$ is $\boxed{[2,\infty)}$.